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Costello was born in Boston, Massachusetts. He served in both the British Royal Air Force and the United States Army Air Corps during World War II. Following his tour of duty, he worked as a foreign news editor for CBS and was a journalist for a number of newspapers and magazines. He studied acting at the Yale School of Drama in New Haven, Connecticut, and at the Old Vic and the University of Birmingham in England.

His first major acting role was starring in an off-Broadway production of August Strindberg's play, The Father. In 1955, he appeared in the Tony Award-winning Broadway production of The Lark, along with Theodore Bikel and Christopher Plummer. While he continued working on the stage in New York, he also performed in live television productions, including a segment of The Alcoa Hour with Richard Kiley.

Costello made his feature film debut as the star of the cult 1958 science fiction film Terror from the Year 5000, which also featured Salome Jens. He then played the supporting role of Captain Harry Black in the 1960 war film The Gallant Hours. Richard Carlyle, Harry Landers, and William Schallert had roles in this film, as well. In addition to acting in the film, Costello also wrote the movie's theme song.

From 1964 through 1966, Costello starred as Peter Ames on the CBS soap opera The Secret Storm. John Colicos and Diana Muldaur was also regulars on that program during Costello's tenure. In 1966, Costello and Graham Jarvis both played witnesses for the prosecution in the Broadway play, The Investigation.